Stat 521A Lecture 18 1

Transcription

1 Stat 521A Lecture 18 1

2 Outline Cts and discrete variables (14.1) Gaussian networks (14.2) Conditional Gaussian networks (14.3) Non-linear Gaussian networks (14.4) Sampling (14.5) 2

3 Hybrid networks A hybrid GM contains discrete and cts variables Except in the case that everything is all discrete or all Gaussian, exact inference is rarely possible The reason is that the basic operations of multiplication, marginalization and conditioning are not closed except for tables and MVNs 3

4 Gaussian networks We can always convert a Gaussian DGM or UGM to an MVN and do exact inference in O(d 2 ) space and O(d 3 ) time However, d can be large (eg 1000x1000 image) We seek methods that exploit the graph structure, that will take O(d w 2 ) space and O(d w 3 ) time, where w is the tree width In cases where w is too large, we can use loopy belief propagation, which takes O(1) space and O(d) time 4

5 Canonical potentials When performing VarElim or ClqTree propagation, we have to represent factors \phi(x). These may not be Gaussians, but can always be represented as exponentials of quadratics 5

6 Operations on canonical potentials Multiplication * = Division 6

7 Operations on canonical potentials Marginalization (requires KYY be pd) Conditioning (Y=y) 7

8 Kalman filter- smoother If you apply the FB algorithm with these new operators, you get the same results as the RTS smoother 8

9 Gaussian LBP If the treewidth is too large, we can pass messages on the original (pairwise) graph We just apply the regular BP rules with the new operators. Once can show this is equivalent to the following: 9

10 Gaussian LBP Thm If LBP converges, then the means are exact, but the variances are too small (overconfident) Thm. A sufficient condition for convergence is that the potentials are pairwise normalizable Any attractive model (all +ve correlations) is pairwise normalizable The method for computing the means is similar to solving a set of linear equations 10

11 Pairwise normalizable Def A pairwise MRF with energies of the form ǫ i (x i ) = d i 0+d i 1x 1 +d i 2x 2 i ǫ ij (x i,x j ) = a i,j 00 +ai,j 01 x i+a ij 10 x j+a ij 11 x ix j +a ij 02 x2 i +a ij 20 x2 j is called pairwise normalizable if d i 2>0, i and ( a ij 02 a ij a ij /2 /2 aij 20 Thm If the MRF is pairwise normalizable, then it defines a valid Gaussian. Sufficient but not necessary eg ) is psd for all i,j May be able to reparameterize the node/ edge potentials to ensure pairwise normalized. 11

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13 Conditional linear Gaussian networks Suppose all discrete nodes only have discrete parents, and all cts nodes either have discrete parents, cts parents, or no parents. Further, assume all cts CPDs have the form p(x=x C=c,D=k)=N(x w T kc,σ 2 k) This is called a CLG network. It is equivalent to a mixture of MVNs, where the distribution over discrete indicators has structure, as does each covariance matrix. We create a canonical factor for each discrete setting of the variables in a clique. 13

14 Inference in CLG networks Thm Inference in CLG networks is NP-hard, even if they are polytrees. Pf (sketch). Consider the network below. When we sum out D 1, p(x 1 ) is a mixture of 2 Gaussians. In general, p(x i ) is a mixture of 2 i Gaussians. 14

15 Weak marginalization To prevent the blowup in the number of mixture components, we can project back to the class of single mixtures at each step, as in EP Prop argmin_q KL(p q) where q is a Gaussian has parameters ( Prop argmin_q KL(p,q) where p is a mixture of Gaussians is a single Gaussian with params M projection 15

16 Weak marginalization 16

17 Canonical vs moment form Weak marginalization is defined in terms of moment form To convert a canonical factor to moment form, we require that it represent a valid joint density This typically requires we pass messages from parents to children. Once we have initialized all factors, they can be converted to moment form. However, division in the backwards pass may cause some variances to become negative! (see Ex ) EP is hairy! 17

18 Strong marginalization By using a constrained elimination order, in which we integrate out before summing out, we can ensure that the upwards pass never needs to perform weak marginalization. Furthermore, one can show that the downwards pass results in exact results for the discrete variables and exact 1 st and 2 nd moments for the cts variables (Lauritzen s strong jtree algorithm) However, the constrained elim order usually results in large discrete cliques, making this often impractical. 18

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20 Non linear dependencies In a linear Gaussian network, the mean is a linear function of its parents. Now assume X i = f(u i, Z i ), where Z i ~ N(0,I) Examples 20

21 Taylor series approx We can linearize f and then fit a Gaussian (basis of the EKF algorithm) Can be bad if f not linear near mu 21

22 M projection using quadrature Best Gaussian approx has these moments Gaussian quadrature computes this integral for any W(z)>0 (here, Gaussian) 22

23 Unscented transform Pass mean and +- std in each dim through transform, and then fit Gaussian to transformed points 23

24 Nonlinear GMs We approximate nonlinear factors by approximating them by Gaussians The above methods require a joint Gaussian factor, not a canonical factor we have to pass messages in topological order, and introduce variables one at a time to use the above tricks Linearization is done relative to current \mu. In EP, we iterate, and re-approximate each factor in the context of its incoming messages, which provides a better approx. to the posterior. Pretty hairy. 24

25 Discrete children, cts parents C -> D arcs are useful eg thermostat turns on/off depending on temperature We can approximate Gaussian * logistic by a Gaussian (variational approx) We can combine these Gaussian factors with the other factors as usual. 25

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27 Sampling Sampling is the easiest way to handle cts and mixed variables Collapsed particles (Rao-Blackwellisation): sample the discretes, integrate out cts analytically. Each particle has a value for D and a Gaussian over C. Good for PF or MCMC. 27

28 Non-parametric BP We can combine sampling and msg passing. We approximate factors/ msgs by samples. Factors are lower dimensional than full joints. Eg hand-pose tracking 28

29 Adaptive discretization We can discretize all the cts variables, then use a method for discrete vars. To increase accuracy, we expand the grid resolution for variables whose posterior entropy is high. Can use such approximations as proposal distributions for MH. 29